An ultrahigh-fidelity 3D holographic display using scattering to homogenize the angular spectrum

A three-dimensional (3D) holographic display (3DHD) can preserve all the volumetric information about an object. However, the poor fidelity of 3DHD constrains its applications. Here, we present an ultrahigh-fidelity 3D holographic display that uses scattering for homogenization of angular spectrum. A scattering medium randomizes the incident photons and homogenizes the angular spectrum distribution. The redistributed field is recorded by a photopolymer film with numerous modulation modes and a half-wavelength scale pixel size. We have experimentally improved the contrast of a focal spot to 6 × 106 and tightened its spatial resolution to 0.5 micrometers, respectively ~300 and 4.4 times better than digital approaches. By exploiting the spatial multiplexing ability of the photopolymer and the transmission channel selection capability of the scattering medium, we have realized a dynamic holographic display of 3D spirals consisting of 20 foci across 1 millimeter × 1 millimeter × 26 millimeters with uniform intensity.


(S4)
According to the Jacobi-Anger expansion, (S5) We only consider N = 0 and 1, (S6) Thus, (S7) Thus, the first item on the right side gives the conjugated phase of the scattered field , i.e., the recorded field is also recovered.

Note S2. Recording and recovering multiply scattered holograms
When the SM and photopolymer remain motionless, if multiple recorded light fields carrying different object information interfere one by one with the reference light field, they will form a superimposed hologram on the photopolymer.Under irradiation, all the recorded light fields will be recovered at once, and the superimposed images of multiple objects will be displayed.The theoretical proof is as follows.
The sum of the holograms' intensities is where .
According to the Jacobi-Anger expansion, (S10) Only consider N = 0 and 1 (S13) Thus we always can find a term of to playback a conjugated scattered light with a phase of .

Note S3. The theoretical maximum peak-to-background ratio (PBR) of focus through scattering medium modulated by the phase-only modulation devices
For simplicity, we assume that each element in the transmission matrix T is drawn from a circular gaussian distribution with mean and .In the cases when T is a unitary matrix such that , can be proved to be , where N is the dimension of the matrix.
We assume the input field is , with only the first element to be nonzero while the rest elements are zero.Also, both photopolymer and LC-SLM have phase only modulation capability.Thus, the peak of the playback field is calculated as (S14) When N is large, the discrete summation can be converted into continuous integration.
(S15) Thus, (S16) Then, the background of the playback field is calculated as Thus, the background intensity is (S18) Here, denotes the ensemble average.Since satisfy a circular gaussian distribution and has a randomly distributed phase, their product also satisfies the same circular gaussian distribution.Thus, we get (S19) Finally, we can calculate the peak-background ratio (PBR) of the focus (S20) The PBR of the recovered focus is proportional to the modulation modes for a holographic device with phase-only modulation.Moreover, the modulation modes of LC-SLM is limited (commonly 2 million modes for commercial high-resolution SLM), thus the PBR of focus reconstructed by the LC-SLM is limited by the modulation modes (Fig. S2).In contrast, since the photopolymer is an analog device, the modulation unit of the photopolymer is much smaller than the wavelength-level speckles.The modulation modes of the photopolymer is more than two orders of magnitude higher than LC-SLM, hence the PBR of the focus is much higher, which is consistent with the results in Fig. 1c.

Note S4. Dynamic 3D holographic display enabled by the memory effect
, (S21) where , L, and are the wavenumber, diffusion thickness, and transport mean free path, respectively.
In dynamic holography, we consider that the deflection between the beam and the SM remains the same, and only the shift Δ changes, so We define , ， where is the beam radius, and then the autocorrelation coefficient of the transmission matrix before and after beam rotation is We performed a simple numerical simulation of the memory effect.When L, , , and the wavelength have values of 258 μm, 14.8 mm which was same as Reference (48), 5 mm, and 532 nm, respectively, the relationship between and is as depicted in Fig. S7.
That is, when the rotation angle Δ deviates by over 1 mrad, the transmission matrix can be considered as irrelevant.Thus, we can display the correct frame image.In our implementation, we utilized the 3D focus scanning module shown in Fig. S5 to generate two focused spots at different locations.The photopolymer was rotated over ~5 mrad using an electric precision rotary stage, and the positions of both foci were recorded with the HAS-3DHD prototype.
During the reconstruction, we captured the intensity distribution of each focus separately with an sCMOS camera (Fig. S10a and b).Fig. S10c displays the lateral intensity profiles of the two foci.It can be observed that after rotating the photopolymer over 5 mrad, the adjacent images could be completely separated.

Note S5. The relationship between the PBR and distance d from the holographic target and scattering medium
The peak-to-background ratio (PBR) of the reconstructed focus in HAS-3DHD is proportional to the ratio of the number of control modes N in the photopolymer to the number of speckles in the reconstructed image M (49, 50).In particular, as the distance d between the recorded object (USAF 1951 resolution chart) and the scattering medium increases, the diameter of the speckles D s will also increase, following the relationship (51) (S24) Here, represents light wavelength, represents the total illuminated area of the SM.Therefore, the number of speckles in the reconstructed image M decreases as the distance d increases, which means the PBR increases with increasing d.The relationship between PBR and d can be theoretically expressed as follows: (S25) Here, represents speckle size at the recorded and reconstructed position; S represents the effected modulation area; K represents a constant related to the modulation capability of the photopolymer for phase modulation, ; . In practical scenarios, due to the influence of various noise factors, the above relationship may not strictly hold as d increases.

Fig. S1 .
Fig. S1.Principle of spatial multiplexing with a photopolymer.(a) Multiple wavefronts are sequentially recorded as holograms on the photopolymer.The holograms are superimposed over each other at the same position.(b) In the absence of scattering medium transmission channel selection, all the recorded wavefronts are reconstructed at the same time, therefore dynamic display cannot be achieved.

Fig. S2 .
Fig. S2.PBRs of a reconstructed focus, using the HAS-3DHD or SLM-based holographic display with different modulation modes, N.

Fig. S4 .
Fig. S4.PBRs of a reconstructed image of a 1951USAF test target, using the HAS-3DHD or SLM-based holographic display at different distances, d.

Fig. S6 .Fig. S7 .
Fig. S6.Normalized intensity profile of the first five foci in the spiral line 3D holographic display.

Fig. S8 .
Fig. S8.Increasing the number of points by recording multiple holograms.(a) Scanning path of the 3D focus scanning module.The numbers and arrows illustrate the sequence of 92 frames.(b) Superposition of frames drawing "CIT" in the image plane.

Fig. S9 .
Fig. S9.Increasing the number of points by recording holograms in different areas of the photopolymer.(a) Translational spatial multiplexing.(b) Rotating spatial multiplexing.The black arrow indicates the direction photopolymer moves or rotates.Masks are added to block stray light in the recording step.

Fig. S10 .
Fig. S10.Reconstructing adjacent images utilizing the memory effect of scattering medium and the multiplexing ability of photopolymer.(a) Reconstructing the foci separately by rotating the photopolymer by 5 mrad.(b) Normalized intensity profiles along the vertical white dashed lines in (a).

Fig. S11 .
Fig. S11.Reconstructing the holographic image at different wavelengths.(a, b) The reconstructed resolution target when the distance d between the target and SM was 20 mm at 640 nm (a) and 532 nm (b).(c) Normalized intensity profiles along the vertical white dashed lines in (a) at 640 nm and 532 nm.

Fig. S12 .
Fig. S12.The images of the photopolymer and diffuser used in the HAS-3DHD experiments.(a) A photographic image of the photopolymer.(b) A schematic of the photopolymer structure, comprising a three-layer stack: a 60±2 µm thick cellulose (TAC) substrate, a 16 ± 2 µm thick light-sensitive photopolymer film, and a ~40 µm thick protective PE cover film, which is removed prior to hologram recording (52).(c) A photographic image of the diffuser.(d) A microscope image of the diffuser.Scale bar, 250 µm.